Pascal's Wager

First published Sat May 2, 1998; substantive revision Tue Nov 6, 2012

“Pascal's Wager” is the name given to an argument due to Blaise Pascal for believing, or for at least taking steps to believe, in God. The name is somewhat misleading, for in a single paragraph of his Pensées. Pascal apparently presents at least three such arguments, each of which might be called a ‘wager’ — it is only the final of these that is traditionally referred to as “Pascal's Wager”. We find in it the extraordinary confluence of several important strands of thought: the justification of theism; probability theory and decision theory, used here for almost the first time in history; pragmatism; voluntarism (the thesis that belief is a matter of the will); and the use of the concept of infinity.

We will begin with some brief stage-setting: some historical background, some of the basics of decision theory, and some of the exegetical problems that the Pensées pose. Then we will follow the text to extract three main arguments. The bulk of the literature addresses the third of these arguments, as will the bulk of our discussion here. Some of the more technical and scholarly aspects of our discussion will be relegated to lengthy footnotes, to which there are links for the interested reader. All quotations are from §233 of Pensées (1910, Trotter translation), the ‘thought’ whose heading is “Infinite—nothing ”.

1. Background

It is important to contrast Pascal's argument with various putative ‘proofs’ of the existence of God that had come before it. Anselm's ontological argument, Aquinas' ‘five ways’, Descartes' ontological and cosmological arguments, and so on, purport to prove that God exists. Pascal is apparently unimpressed by such attempted justifications of theism: “Endeavour. to convince yourself, not by increase of proofs of God. ” Indeed, he concedes that “we do not know if He is. ”. Pascal's project, then, is radically different: he seeks to provide prudential reasons for believing in God. To put it crudely, we should wager that God exists because it is the best bet. Ryan 1994 finds precursors to this line of reasoning in the writings of Plato, Arnobius, Lactantius, and others; we might add Ghazali to his list — see Palacios 1920. But what is distinctive is Pascal's explicitly decision theoretic formulation of the reasoning. In fact, Hacking 1975 describes the Wager as “the first well-understood contribution to decision theory” (viii). Thus, we should pause briefly to review some of the basics of that theory.

In any decision problem, the way the world is, and what an agent does, together determine an outcome for the agent. We may assign utilities to such outcomes, numbers that represent the degree to which the agent values them. It is typical to present these numbers in a decision matrix, with the columns corresponding to the various relevant states of the world, and the rows corresponding to the various possible actions that the agent can perform.

In decisions under uncertainty. nothing more is given — in particular, the agent does not assign subjective probabilities to the states of the world. Still, sometimes rationality dictates a unique decision nonetheless. Consider, for example, a case that will be particularly relevant here. Suppose that you have two possible actions, A1 and A2, and the worst outcome associated with A1 is at least as good as the best outcome associated with A2; suppose also that in at least one state of the world, A1's outcome is strictly better than A2's. Let us say in that case that A1 superdominates A2. Then rationality seems to require you to perform A1. [1 ]

In decisions under risk. the agent assigns subjective probabilities to the various states of the world. Assume that the states of the world are independent of what the agent does. A figure of merit called the expected utility. or the expectation of a given action can be calculated by a simple formula: for each state, multiply the utility that the action produces in that state by the state's probability; then, add these numbers. According to decision theory, rationality requires you to perform the action of maximum expected utility (if there is one).

Example. Suppose that the utility of money is linear in number of dollars: you value money at exactly its face value. Suppose that you have the option of paying a dollar to play a game in which there is

an equal chance of returning nothing, and returning three dollars. The expectation of the game itself is

0×(1/2) + 3×(1/2) = 1.5,

so the expectation of paying a dollar for certain, then playing, is

-1 + 1.5 = 0.5.

This exceeds the expectation of not playing (namely 0), so you should play. On the other hand, if the game gave an equal chance of returning nothing, and returning two dollars, then its expectation would be:

0×(1/2) + 2×(1/2) = 1.

Then consistent with decision theory, you could either pay the dollar to play, or refuse to play, for either way your overall expectation would be 0.

Considerations such as these will play a crucial role in Pascal's arguments. It should be admitted that there are certain exegetical problems in presenting these arguments. Pascal never finished the Pensées. but rather left them in the form of notes of various sizes pinned together. Hacking 1972 describes the “Infinite—nothing” as consisting of “two pieces of paper covered on both sides by handwriting going in all directions, full of erasures, corrections, insertions, and afterthoughts” (24). [2 ] This may explain why certain passages are notoriously difficult to interpret, as we will see. Furthermore, our formulation of the arguments in the parlance of modern Bayesian decision theory might appear somewhat anachronistic. For example, Pascal did not distinguish between what we would now call objective and subjective probability, although it is clear that it is the latter that is relevant to his arguments. To some extent, “Pascal's Wager” now has a life of its own, and our presentation of it here is perfectly standard. Still, we will closely follow Pascal's text, supporting our reading of his arguments as much as possible.

There is the further problem of dividing the Infinite-nothing into separate arguments. We will locate three arguments that each conclude that rationality requires you to wager for God, although they interleave in the text. [3 ] Finally, there is some disagreement over just what “wagering for God” involves — is it believing in God, or merely trying to? We will conclude with a discussion of what Pascal meant by this.

2. The Argument from Superdominance

Pascal maintains that we are incapable of knowing whether God exists or not, yet we must “wager” one way or the other. Reason cannot settle which way we should incline, but a consideration of the relevant outcomes supposedly can. Here is the first key passage:

“God is, or He is not.” But to which side shall we incline? Reason can decide nothing here. There is an infinite chaos which separated us. A game is being played at the extremity of this infinite distance where heads or tails will turn up. Which will you choose then? Let us see. Since you must choose, let us see which interests you least. You have two things to lose, the true and the good; and two things to stake, your reason and your will, your knowledge and your happiness; and your nature has two things to shun, error and misery. Your reason is no more shocked in choosing one rather than the other, since you must of necessity choose. But your happiness? Let us weigh the gain and the loss in wagering that God is. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.

There are exegetical problems already here, partly because Pascal appears to contradict himself. He speaks of “the true” as something that you can “lose”, and “error” as something “to shun”. Yet he goes on to claim that if you lose the wager that God is, then “you lose nothing”. Surely in that case you “lose the true”, which is just to say that you have made an error. Pascal believes, of course, that the existence of God is “the true” — but that is not something that he can appeal to in this argument. Moreover, it is not because “you must of necessity choose” that “your reason is no more shocked in choosing one rather than the other”. Rather, by Pascal's own account, it is because “[r]eason can decide nothing here”. (If it could, then it might well be shocked — namely, if you chose in a way contrary to it.)

Following McClennen 1994, Pascal's argument seems to be best captured as presenting the following decision matrix: